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Fractal uncertainty for discrete 2D Cantor sets

Alex Cohen

TL;DR

This work establishes a fractal uncertainty principle (FUP) for discrete two-dimensional Cantor sets in $\mathbb{Z}_N^2$ and identifies the exact obstruction: the FUP holds if the limiting Cantor sets $\mathbf X,\mathbf Y$ do not contain a pair of orthogonal lines. The proof combines a submultiplicativity/induction-on-scales framework with a sharp arithmetic input: a quantitative form of Lang’s conjecture due to Ruppert and Beukers & Smyth, guaranteeing that low-degree trigonometric polynomials cannot vanish on too many cyclotomic points unless a line is present. A central device is a multiplier built from a polynomial $F^*$ of degree $<200|S|^{1/2}$ that concentrates mass along a line when $S$ is the support of $f$, enabling a 2D FUP after ruling out line obstructions. The results yield spectral-gap implications for two-dimensional open baker maps and connect fractal geometry, number theory, and quantum chaos with a rigorous discrete-model perspective. The framework sharpens the 2D analogue of known 1D FUP results and provides a precise criterion for when FUP fails due to linear obstructions.

Abstract

We prove that a self-similar Cantor set in $\mathbb{Z}_N \times \mathbb{Z}_N$ has a fractal uncertainty principle if and only if it does not contain a pair of orthogonal lines. The key ingredient in our proof is a quantitative form of Lang's conjecture in number theory due to Ruppert and Beukers & Smyth. Our theorem answers a question of Dyatlov and has applications to open quantum maps.

Fractal uncertainty for discrete 2D Cantor sets

TL;DR

This work establishes a fractal uncertainty principle (FUP) for discrete two-dimensional Cantor sets in and identifies the exact obstruction: the FUP holds if the limiting Cantor sets do not contain a pair of orthogonal lines. The proof combines a submultiplicativity/induction-on-scales framework with a sharp arithmetic input: a quantitative form of Lang’s conjecture due to Ruppert and Beukers & Smyth, guaranteeing that low-degree trigonometric polynomials cannot vanish on too many cyclotomic points unless a line is present. A central device is a multiplier built from a polynomial of degree that concentrates mass along a line when is the support of , enabling a 2D FUP after ruling out line obstructions. The results yield spectral-gap implications for two-dimensional open baker maps and connect fractal geometry, number theory, and quantum chaos with a rigorous discrete-model perspective. The framework sharpens the 2D analogue of known 1D FUP results and provides a precise criterion for when FUP fails due to linear obstructions.

Abstract

We prove that a self-similar Cantor set in has a fractal uncertainty principle if and only if it does not contain a pair of orthogonal lines. The key ingredient in our proof is a quantitative form of Lang's conjecture in number theory due to Ruppert and Beukers & Smyth. Our theorem answers a question of Dyatlov and has applications to open quantum maps.
Paper Structure (27 sections, 26 theorems, 161 equations, 5 figures)

This paper contains 27 sections, 26 theorems, 161 equations, 5 figures.

Table of Contents

  1. Introduction
  2. One dimensional fractal uncertainty
  3. Two dimensional fractal uncertainty
  4. Sketch of the argument
  5. An application to quantum chaos
  6. Organization
  7. The 1D argument
  8. The 2D argument
  9. Discretization of fractal sets
  10. Some useful lemmas on lines
  11. Proof of Theorem \ref{['thm:2D_FUP']} assuming Lemma \ref{['lem:main_lemma']}
  12. Proof of the main Lemma
  13. Loose ends
  14. Submultiplicativity
  15. Theorem \ref{['thm:2D_FUP']} is sharp
  16. ...and 12 more sections

Key Result

Theorem 1

For all alphabets $\mathcal{A}, \mathcal{B} \subsetneq \mathbb{Z}_M$, the estimate holds for some $\beta > 0$.

Figures (5)

  • Figure 1: Cantor sets can contain lines that aren't horizontal, vertical, or diagonal, but they are less stable
  • Figure 2: Diagram of the 1D argument
  • Figure 3: Visualization of Lemma \ref{['lem:main_lemma']}
  • Figure 4: The two cases in Proposition \ref{['prop:main_FUP_prop']} obtain contradictions in different ways.
  • Figure 5: Cases in the proof of Lemma \ref{['lem:polynomials_through_sets']}

Theorems & Definitions (50)

  • Theorem 1: 1D FUP DyatlovJinBakersMaps*Theorem 2
  • Theorem 2: 2D FUP
  • Proposition 1
  • Remark 1
  • Corollary 1
  • Proposition 2: DyatlovJinBakersMaps*Proposition 2.6
  • Theorem 3: DyatlovJinBakersMaps*Theorem 1
  • Theorem 4
  • Proposition 3
  • proof
  • ...and 40 more